A:There is a lot of discussion on the web on using hypergeometric distributions for solving these kind of problems. The hypergeometric distribution is just a big word for something fairly simple. Here is the wikipedia explanation for it (link).
But an easier "smarter" way to solve this puzzle (along with ones that fit this framework) is to work with expectations and indicator random variables. Indicator random variables are like on/off switches. They are 1 under certain conditions, 0 otherwise.

Let us assume that the deck of cards (52 total) is as follows, it has 4 aces and all others are labelled X. Let \(Z_i\) represent the indicator variable for a card in position \(i\) with value set as 1 when all aces are in behind it, 0 otherwise.

The total number of draws will then be

$$ N = \sum_{i=1}^{48} Z_{i}$$

Now consider the first draw. The probability that all the aces in the deck is behind it is \(\frac{1}{5}\). This step is crucial to understand. Do you see why it is \(\frac{1}{5}\)? See the figure below. There are 5 scenarios that can play out in terms after the first draw of how the 4 aces position themselves with respect to the current card being pulled AND they are all equally likely.

Only the top row is the one that causes us to draw more cards. This implies that the expectation of \(Z_i\) is always \(\frac{1}{5}\). We run the sum to 48, because after we draw 48 cards there will only be aces left. Thus the sum above yields \(\frac{48}{5} = 9.6\).
Notice when we said that the first row was the only favourable one, it implicitly implied that we are NOT drawing an Ace. Thus the actual count of cards drawn would be 1 + 9.6 = 10.6.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.